COS3761 Formal Logic III Exam 2026 |
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📚 Section 1: Propositional Logic – Syntax & Semantics (Questions 1–40)
1. Which of the following is a well-formed formula (WFF) in propositional logic?
A) P ∧ Q ∨
B) (P → Q) ∧ R
C) P → Q ∧
D) ∨ P ∧ Q
Answer: B
Explanation: A well-formed formula must follow the syntactic rules of propositional logic. (P
→ Q) ∧ R is correctly formed with binary connectives between appropriate subformulas.
Options A, C, and D have connectives placed incorrectly or lack proper parentheses.
, 2. The formula ¬(P ∧ Q) is logically equivalent to:
A) ¬P ∧ ¬Q
B) ¬P ∨ ¬Q
C) P → ¬Q
D) ¬P → ¬Q
Answer: B
Explanation: ¬(P ∧ Q) ≡ ¬P ∨ ¬Q by De Morgan's Law. This is one of the fundamental
logical equivalences. De Morgan's Laws state that the negation of a conjunction is the
disjunction of the negations.
3. Which of the following is a tautology?
A) P ∧ ¬P
B) P ∨ ¬P
C) P → ¬P
D) P ∧ Q
Answer: B
Explanation: P ∨ ¬P is the law of excluded middle and is always true regardless of the truth
value of P. P ∧ ¬P is a contradiction, P → ¬P is not always true, and P ∧ Q is contingent.
, 4. The implication P → Q is false only when:
A) P is true and Q is true
B) P is false and Q is true
C) P is true and Q is false
D) P is false and Q is false
Answer: C
Explanation: An implication P → Q is false only in the case where the antecedent P is true
and the consequent Q is false. In all other cases, the implication is true.
5. The converse of the implication P → Q is:
A) Q → P
B) ¬Q → ¬P
C) ¬P → ¬Q
D) P ↔ Q
Answer: A
Explanation: The converse of P → Q is Q → P. The contrapositive is ¬Q → ¬P, and the
inverse is ¬P → ¬Q. These are distinct implications and are not logically equivalent.
6. The contrapositive of P → Q is:
A) Q → P
, B) ¬Q → ¬P
C) ¬P → ¬Q
D) P ↔ Q
Answer: B
Explanation: The contrapositive of P → Q is ¬Q → ¬P. The implication and its
contrapositive are logically equivalent (P → Q) ≡ (¬Q → ¬P). This is an important logical
equivalence used in proofs.
7. Which of the following logical equivalences is correct?
A) P → Q ≡ ¬P ∧ Q
B) P → Q ≡ ¬P ∨ Q
C) P → Q ≡ P ∨ ¬Q
D) P → Q ≡ ¬Q → ¬P
Answer: B
Explanation: The correct equivalence is P → Q ≡ ¬P ∨ Q. This is known as the implication
law or material implication. Option D is also true (contrapositive), but the question asks for
the correct equivalence.
8. A truth table for a formula with 3 propositional variables has how many rows?
A) 4
Practice Questions with Answers & Detailed
Explanations | UNISA | Latest Update |
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📚 Section 1: Propositional Logic – Syntax & Semantics (Questions 1–40)
1. Which of the following is a well-formed formula (WFF) in propositional logic?
A) P ∧ Q ∨
B) (P → Q) ∧ R
C) P → Q ∧
D) ∨ P ∧ Q
Answer: B
Explanation: A well-formed formula must follow the syntactic rules of propositional logic. (P
→ Q) ∧ R is correctly formed with binary connectives between appropriate subformulas.
Options A, C, and D have connectives placed incorrectly or lack proper parentheses.
, 2. The formula ¬(P ∧ Q) is logically equivalent to:
A) ¬P ∧ ¬Q
B) ¬P ∨ ¬Q
C) P → ¬Q
D) ¬P → ¬Q
Answer: B
Explanation: ¬(P ∧ Q) ≡ ¬P ∨ ¬Q by De Morgan's Law. This is one of the fundamental
logical equivalences. De Morgan's Laws state that the negation of a conjunction is the
disjunction of the negations.
3. Which of the following is a tautology?
A) P ∧ ¬P
B) P ∨ ¬P
C) P → ¬P
D) P ∧ Q
Answer: B
Explanation: P ∨ ¬P is the law of excluded middle and is always true regardless of the truth
value of P. P ∧ ¬P is a contradiction, P → ¬P is not always true, and P ∧ Q is contingent.
, 4. The implication P → Q is false only when:
A) P is true and Q is true
B) P is false and Q is true
C) P is true and Q is false
D) P is false and Q is false
Answer: C
Explanation: An implication P → Q is false only in the case where the antecedent P is true
and the consequent Q is false. In all other cases, the implication is true.
5. The converse of the implication P → Q is:
A) Q → P
B) ¬Q → ¬P
C) ¬P → ¬Q
D) P ↔ Q
Answer: A
Explanation: The converse of P → Q is Q → P. The contrapositive is ¬Q → ¬P, and the
inverse is ¬P → ¬Q. These are distinct implications and are not logically equivalent.
6. The contrapositive of P → Q is:
A) Q → P
, B) ¬Q → ¬P
C) ¬P → ¬Q
D) P ↔ Q
Answer: B
Explanation: The contrapositive of P → Q is ¬Q → ¬P. The implication and its
contrapositive are logically equivalent (P → Q) ≡ (¬Q → ¬P). This is an important logical
equivalence used in proofs.
7. Which of the following logical equivalences is correct?
A) P → Q ≡ ¬P ∧ Q
B) P → Q ≡ ¬P ∨ Q
C) P → Q ≡ P ∨ ¬Q
D) P → Q ≡ ¬Q → ¬P
Answer: B
Explanation: The correct equivalence is P → Q ≡ ¬P ∨ Q. This is known as the implication
law or material implication. Option D is also true (contrapositive), but the question asks for
the correct equivalence.
8. A truth table for a formula with 3 propositional variables has how many rows?
A) 4
